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author | Karl Berry <karl@freefriends.org> | 2006-01-09 00:49:07 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2006-01-09 00:49:07 +0000 |
commit | 007f67a693e4d031fd3d792df8e4d5f43e2cb2e7 (patch) | |
tree | 90d17e00e572ecb1e24764b6f29c80e098b08d29 /Master/texmf-dist/doc/latex/algorithms/algorithms.tex | |
parent | 950209b26f70aa87ed07c54f82a95b6f03b7c3a0 (diff) |
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diff --git a/Master/texmf-dist/doc/latex/algorithms/algorithms.tex b/Master/texmf-dist/doc/latex/algorithms/algorithms.tex new file mode 100644 index 00000000000..6affdad8b7c --- /dev/null +++ b/Master/texmf-dist/doc/latex/algorithms/algorithms.tex @@ -0,0 +1,728 @@ +% ALGORITHMS for LaTeX version 2e +% +% Current maintainer: Rogério Brito <rbrito@ime.usp.br> +% Previous maintainer: Peter Williams <pwil3058@bigpond.net.au> +% +% This document file is free software; you can redistribute it and/or +% modify it under the terms of the GNU Lesser General Public License as +% published by the Free Software Foundation; either version 2 of the +% License, or (at your option) any later version. +% +% This document file is distributed in the hope that it will be useful, but +% WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser +% General Public License for more details. +% +% You should have received a copy of the GNU Lesser General Public License +% along with this document file; if not, write to the Free Software +% Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, +% USA. +% +\documentclass{article} + +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{lmodern} +\usepackage[pdfstartview={FitH}]{hyperref} +\usepackage{algorithm} +\usepackage{algorithmic} + +\title{Algorithms} +\author{Rogério Brito\\ + \href{mailto:rbrito@ime.usp.br}{rbrito@ime.usp.br}\thanks{Sincere thanks + go to the original maintainer of this package, Peter Williams, and for + being kind enough to allow me to continue with his quite useful work.}} + +\newcommand{\keyword}[1]{\texttt{#1}} +\newcommand{\filename}[1]{\texttt{#1}} +\newcommand{\latcom}[1]{\texttt{$\backslash$#1}} + +\setcounter{tocdepth}{2} + +\begin{document} +\maketitle +\tableofcontents +%\listofalgorithms + +\section{Introduction} + +This package provides two environments, \keyword{algorithmic} and +\keyword{algorithm}, which are designed to be used together but may, +depending on the necessities of the author, be used separately. + +The \keyword{algorithmic} environment provides an environment for +describing algorithms and the \keyword{algorithm} environment provides a +``float'' wrapper for algorithms (implemented using \keyword{algorithmic} +or some other method at the author's option). The reason that two +environments are provided is to allow the author maximum flexibility. + +This work may be distributed and/or modified under the conditions of the +GNU Lesser General Public License as published by the Free Software +Foundation (see the file \filename{COPYING} included in this package). +Currently, this package consists of three files: \filename{algorithm.sty}, +\filename{algorithmic.sty} and \filename{algorithms.tex} (the source of +this document). This is likely to change in the near future. + +\section{The \keyword{algorithmic} Environment} +\label{sec:algorithmic-envir} +Within an \keyword{algorithmic} a number of commands for typesetting +popular algorithmic constructs are available. +In general, the commands provided can be arbitrarily nested to +describe quite complex algorithms. +An optional argument to the \verb+\begin{algorithmic}+ statement can be +used to turn on line numbering by giving a positive integer indicating the +required frequency of line numbering. +For example, \verb+\begin{algorithmic}[5]+ would cause every fifth line to +be numbered. + +\subsection{The Simple Statement} + +The simple statement takes the form +\begin{verbatim} +\STATE <text> +\end{verbatim} +and is used for simple statements, e.g. +\begin{verbatim} +\begin{algorithmic} +\STATE $S \leftarrow 0$ +\end{algorithmic} +\end{verbatim} +would produce +\begin{algorithmic} + \STATE $S \leftarrow 0$ +\end{algorithmic} +and with line numbering selected for every line using +\begin{verbatim} +\begin{algorithmic}[1] +\STATE $S \leftarrow 0$ +\end{algorithmic} +\end{verbatim} +would produce +\begin{algorithmic}[1] + \STATE $S \leftarrow 0$ +\end{algorithmic} +For users of earlier versions of \keyword{algorithmic} this construct is +a cause of an incompatibility. +In the earlier version, instead of starting simple statements with the +\verb+\STATE+ +command, simple statements were entered as free text and terminated with +\verb+\\+ command. +Unfortunately, this simpler method failed to survive the modifications +necessary for statement numbering. +However, the \verb+\\+ command can still be used to force a line break within +a simple statement. + +\subsection{The \emph{if-then-else} Construct} + +The \emph{if-then-else} construct takes the forms. +\begin{verbatim} +\IF{<condition>} <text> \ENDIF +\IF{<condition>} <text1> \ELSE <text2> \ENDIF +\IF{<condition1>} <text1> \ELSIF{<condition2>} <text2> \ELSE <text3> \ENDIF +\end{verbatim} +In the third of these forms there is no limit placed on the number +of \verb+\ELSIF{<C>}+ that may be used. +For example, +\begin{verbatim} +\begin{algorithmic} +\IF{some condition is true} +\STATE do some processing +\ELSIF{some other condition is true} +\STATE do some different processing +\ELSIF{some even more bizarre condition is met} +\STATE do something else +\ELSE +\STATE do the default actions +\ENDIF +\end{algorithmic} +\end{verbatim} +would produce +\begin{algorithmic} + \IF{some condition is true} + \STATE do some processing + \ELSIF{some other condition is true} + \STATE do some different processing + \ELSIF{some even more bizarre condition is met} + \STATE do something else + \ELSE + \STATE do the default actions + \ENDIF +\end{algorithmic} +with appropriate indentations. + +\subsection{The \emph{for} Loop} + +The \emph{for} loop takes the forms. +\begin{verbatim} +\FOR{<condition>} <text> \ENDFOR +\FORALL{<condition>} <text> \ENDFOR +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\FOR{$i=0$ to $10$} +\STATE carry out some processing +\ENDFOR +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \FOR{$i=0$ to $10$} + \STATE carry out some processing + \ENDFOR +\end{algorithmic} +and +\begin{verbatim} +\begin{algorithmic}[1] +\FORALL{$i$ such that $0\leq i\leq 10$} +\STATE carry out some processing +\ENDFOR +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic}[1] + \FORALL{$i$ such that $0\leq i\leq 10$} + \STATE carry out some processing + \ENDFOR +\end{algorithmic} + +\subsection{The \emph{while} Loop} + +The \emph{while} loop takes the form. +\begin{verbatim} +\WHILE{<condition>} <text> \ENDWHILE +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\WHILE{some condition holds} +\STATE carry out some processing +\ENDWHILE +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \WHILE{some condition holds} + \STATE carry out some processing + \ENDWHILE +\end{algorithmic} + +\subsection{The \emph{repeat-until} Loop} + +The \emph{repeat-until} loop takes the form. +\begin{verbatim} +\REPEAT <text> \UNTIL{<condition>} +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\REPEAT +\STATE carry out some processing +\UNTIL{some condition is met} +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \REPEAT + \STATE carry out some processing + \UNTIL{some condition is met} +\end{algorithmic} + +\subsection{The Infinite Loop} + +The infinite loop takes the form. +\begin{verbatim} +\LOOP <text> \ENDLOOP +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\LOOP +\STATE this processing will be repeated forever +\ENDLOOP +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \LOOP + \STATE this processing will be repeated forever + \ENDLOOP +\end{algorithmic} + +\subsection{The Precondition} + +The precondition (that must be met if an algorithm is to correctly +execute) takes the form: +\begin{verbatim} +\REQUIRE <text> +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\REQUIRE $x \neq 0$ and $n \geq 0$ +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \REQUIRE $x \neq 0$ and $n \geq 0$ +\end{algorithmic} + +\subsection{The Postcondition} + +The postcondition (that must be met after an algorithm has correctly +executed) takes the form: +\begin{verbatim} +\ENSURE <text> +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\ENSURE $x \neq 0$ and $n \geq 0$ +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \ENSURE $x \neq 0$ and $n \geq 0$ +\end{algorithmic} + +\subsection{Returning Values} + +The \keyword{algorithmic} environment offers a special statement for +explicitly returning values in algorithms. It has the syntax: +\begin{verbatim} +\RETURN <text> +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\RETURN $(x+y)/2$ +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \RETURN $(x+y)/2$ +\end{algorithmic} + +\subsection{Printing Messages} + +Another feature of the \keyword{algorithmic} environment is that it +currently provides a standard way of printing values (which is an operation +used enough to merit its own keyword). It has the syntax: +\begin{verbatim} +\PRINT <text> +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\PRINT \texttt{``Hello, World!''} +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \PRINT \texttt{``Hello, World!''} +\end{algorithmic} + +\subsection{Comments} + +Comments may be inserted at most points in an algorithm using the form: +\begin{verbatim} +\COMMENT{<text>} +\end{verbatim} +For example, +\begin{verbatim} +\begin{algorithmic} +\STATE do something \COMMENT{this is a comment} +\end{algorithmic} +\end{verbatim} +produces +\begin{algorithmic} + \STATE do something \COMMENT{this is a comment} +\end{algorithmic} +Because the mechanisms used to build the various algorithmic structures +make it difficult to use the above mechanism for placing comments at the +end of the first line of a construct, the commands \verb+\IF+, +\verb+\ELSIF+, \verb+\ELSE+, \verb+\WHILE+, \verb+\FOR+, +\verb+\FORALL+, \verb+\REPEAT+ +and \verb+\LOOP+ all take an optional argument which will be treated +as a comment to be placed at the end of the line on which they appear. +For example, +\begin{algorithmic} + \REPEAT[this is comment number one] + \IF[this is comment number two]{condition one is met} + \STATE do something + \ELSIF[this is comment number three]{condition two is met} + \STATE do something else + \ELSE[this is comment number four] + \STATE do nothing + \ENDIF + \UNTIL{hell freezes over} +\end{algorithmic} + +\subsection{An Example} + +The following example demonstrates the use of the \keyword{algorithmic} +environment to describe a complete algorithm. The following input +\begin{verbatim} +\begin{algorithmic} +\REQUIRE $n \geq 0$ +\ENSURE $y = x^n$ +\STATE $y \Leftarrow 1$ +\STATE $X \Leftarrow x$ +\STATE $N \Leftarrow n$ +\WHILE{$N \neq 0$} +\IF{$N$ is even} +\STATE $X \Leftarrow X \times X$ +\STATE $N \Leftarrow N / 2$ +\ELSE[$N$ is odd] +\STATE $y \Leftarrow y \times X$ +\STATE $N \Leftarrow N - 1$ +\ENDIF +\ENDWHILE +\end{algorithmic} +\end{verbatim} +will produce +\begin{algorithmic} + \REQUIRE $n \geq 0$ + \ENSURE $y = x^n$ + \STATE $y \Leftarrow 1$ + \STATE $X \Leftarrow x$ + \STATE $N \Leftarrow n$ + \WHILE{$N \neq 0$} + \IF{$N$ is even} + \STATE $X \Leftarrow X \times X$ + \STATE $N \Leftarrow N / 2$ + \ELSE[$N$ is odd] + \STATE $y \Leftarrow y \times X$ + \STATE $N \Leftarrow N - 1$ + \ENDIF + \ENDWHILE +\end{algorithmic} +which is an algorithm for finding the value of a number taken to a +non-negative power. + +\subsection{Options and Customization} + +There is a single option, \keyword{noend}\label{kwd:noend} that may be +invoked when the \texttt{algorithmic} package is loaded. With this option +invoked the \emph{end} statements are omitted in the output. This allows +space to be saved in the output document when this is an issue. + +\subsubsection{Changing Indentation} +\label{sec:changing-indentation} +In the spirit of saving vertical space (which is especially important when +submitting a paper for a journal, where space is frequently limited for +authors), the \keyword{algorithmic} environment offers, beginning with the +version released in 2005-05-08, a way to control the amount of indentation +that is used by a given algorithm. + +The amount of indentation to be used is given by the command +\begin{verbatim} +\algsetup{indent=lenght} +\end{verbatim} +where \emph{length} is any valid length used by \TeX. The default value of +the indentation used by the \keyword{algorithmic} environment is $1$ em +(for ``backward compatibility reasons''), but a value of $2$ em or more is +recommended, depending on the publication. For example, the snippet +\begin{verbatim} +\algsetup{indent=2em} +\begin{algorithmic}[1] + \STATE $a \Leftarrow 1$ + \IF{$a$ is even} + \PRINT ``$a$ is even'' + \ELSE + \PRINT ``$a$ is odd'' +\end{algorithmic} +\end{verbatim} +produces +\algsetup{indent=2em} +\begin{algorithmic}[1] + \STATE $a \Leftarrow 1$ + \IF{$a$ is even} + \PRINT ``$a$ is even'' + \ELSE + \PRINT ``$a$ is odd'' + \ENDIF +\end{algorithmic} +while +\begin{verbatim} +\algsetup{indent=5em} +\begin{algorithmic}[1] + \STATE $a \Leftarrow 1$ + \IF{$a$ is even} + \PRINT ``$a$ is even'' + \ELSE + \PRINT ``$a$ is odd'' +\end{algorithmic} +\end{verbatim} +would produce +\algsetup{indent=5em} +\begin{algorithmic}[1] + \STATE $a \Leftarrow 1$ + \IF{$a$ is even} + \PRINT ``$a$ is even'' + \ELSE + \PRINT ``$a$ is odd'' + \ENDIF +\end{algorithmic} +\algsetup{indent=1em} + +The intended use of this option is to allow the author to omit the +\emph{end} (see Section~\ref{kwd:noend} for details) statements without +loosing readability, by increasing the amount of indentation to a suitable +level. + +\subsubsection{Changing Line Numbering} + +As mentioned in Section~\ref{sec:algorithmic-envir} and illustrated in +Section~\ref{sec:changing-indentation}, \keyword{algorithms} already +provides you with the possibility of numbering lines. + +Starting with the version released in 2005-07-05, you can now change two +aspects of line numbering: the size of the line numbers (which, by default, +is \verb+\footnotesize+) and the delimiter used to separate the line number +from the code (which, by default, is \verb+:+, i.e., a colon). + +You can change the size of the line numbers using the command: +\begin{verbatim} +\algsetup{linenosize=size} +\end{verbatim} +where \emph{size} is any of the various commands provided by \LaTeX\ to +change the size of the font to be used. Among others, useful values are +\verb+\tiny+, \verb+\scriptsize+, \verb+\footnotesize+ and \verb+\small+. +Please see the complete list of sizes in your \LaTeX\ documentation. + +As another frequently requested feature, you can change the delimiter used +with the line numbers by issuing the command: +\begin{verbatim} +\algsetup{linenodelimiter=delimiter} +\end{verbatim} +where \emph{delimiter} is any ``well-formed'' string, including the empty +string. With this command, you can change the colon to a period (\verb+.+) +by issuing the command +\begin{verbatim} +\algsetup{linenodelimiter=.} +\end{verbatim} +or even omit the delimiter, by specifying the empty string or a space +(\verb+\ +), whatever seems best for your document. + +As an example of such commands, the code produced by +\begin{verbatim} +\algsetup{linenosize=\small, + linenodelimiter=.} +\begin{algorithmic}[1] + \STATE $i \leftarrow 10$ + \RETURN $i$ +\end{algorithmic} +\end{verbatim} +would be something like +\algsetup{linenosize=\small, + linenodelimiter=.} +\begin{algorithmic}[1] + \STATE $i \leftarrow 10$ + \RETURN $i$ +\end{algorithmic} +\algsetup{linenosize=\footnotesize, + linenodelimiter=:} + +\subsubsection{Customization} + +In order to facilitate the use of this package with foreign languages, all +of the words in the output are produced via redefinable macro commands. +The default definitions of these macros are: +\begin{verbatim} +\newcommand{\algorithmicrequire}{\textbf{Require:}} +\newcommand{\algorithmicensure}{\textbf{Ensure:}} +\newcommand{\algorithmicend}{\textbf{end}} +\newcommand{\algorithmicif}{\textbf{if}} +\newcommand{\algorithmicthen}{\textbf{then}} +\newcommand{\algorithmicelse}{\textbf{else}} +\newcommand{\algorithmicelsif}{\algorithmicelse\ \algorithmicif} +\newcommand{\algorithmicendif}{\algorithmicend\ \algorithmicif} +\newcommand{\algorithmicfor}{\textbf{for}} +\newcommand{\algorithmicforall}{\textbf{for all}} +\newcommand{\algorithmicdo}{\textbf{do}} +\newcommand{\algorithmicendfor}{\algorithmicend\ \algorithmicfor} +\newcommand{\algorithmicwhile}{\textbf{while}} +\newcommand{\algorithmicendwhile}{\algorithmicend\ \algorithmicwhile} +\newcommand{\algorithmicloop}{\textbf{loop}} +\newcommand{\algorithmicendloop}{\algorithmicend\ \algorithmicloop} +\newcommand{\algorithmicrepeat}{\textbf{repeat}} +\newcommand{\algorithmicuntil}{\textbf{until}} +\newcommand{\algorithmicprint}{\textbf{print}} +\newcommand{\algorithmicreturn}{\textbf{return}} +\end{verbatim} + +In addition, the formatting of comments is implemented via a single +argument command macro which may also be redefined. +The default definition is +\begin{verbatim} +\newcommand{\algorithmiccomment}[1]{\{#1\}} +\end{verbatim} +and another option that may be interesting for users familiar with C-like +languages is to redefine the comments to be +\begin{verbatim} +\renewcommand{\algorithmiccomment}[1]{//#1} +\end{verbatim} + +Comments produced this way would be like this: +\renewcommand{\algorithmiccomment}[1]{//#1} +\begin{algorithmic} + \STATE $i \leftarrow i +1$ \COMMENT{Increments $i$} +\end{algorithmic} +This second way to present comments may become the default in a future +version of the package. + +\section{The \keyword{algorithm} Environment} + +\subsection{General} +\begin{algorithm} +\caption{Calculate $y = x^n$} +\label{alg1} +\begin{algorithmic} +\REQUIRE $n \geq 0 \vee x \neq 0$ +\ENSURE $y = x^n$ +\STATE $y \Leftarrow 1$ +\IF{$n < 0$} +\STATE $X \Leftarrow 1 / x$ +\STATE $N \Leftarrow -n$ +\ELSE +\STATE $X \Leftarrow x$ +\STATE $N \Leftarrow n$ +\ENDIF +\WHILE{$N \neq 0$} +\IF{$N$ is even} +\STATE $X \Leftarrow X \times X$ +\STATE $N \Leftarrow N / 2$ +\ELSE[$N$ is odd] +\STATE $y \Leftarrow y \times X$ +\STATE $N \Leftarrow N - 1$ +\ENDIF +\ENDWHILE +\end{algorithmic} +\end{algorithm} + +When placed within the text without being encapsulated in a floating +environment \texttt{algorithmic} environments may be split over a page +boundary greatly detracting from their appearance. +In addition, it is useful to have algorithms numbered for reference +and for lists of algorithms to be appended to the list of contents. +The \texttt{algorithm} environment is meant to address these concerns +by providing a floating environment for algorithms. +For example, the input text +\begin{verbatim} +\begin{algorithm} +\caption{Calculate $y = x^n$} +\label{alg1} +\begin{algorithmic} +\REQUIRE $n \geq 0 \vee x \neq 0$ +\ENSURE $y = x^n$ +\STATE $y \Leftarrow 1$ +\IF{$n < 0$} +\STATE $X \Leftarrow 1 / x$ +\STATE $N \Leftarrow -n$ +\ELSE +\STATE $X \Leftarrow x$ +\STATE $N \Leftarrow n$ +\ENDIF +\WHILE{$N \neq 0$} +\IF{$N$ is even} +\STATE $X \Leftarrow X \times X$ +\STATE $N \Leftarrow N / 2$ +\ELSE[$N$ is odd] +\STATE $y \Leftarrow y \times X$ +\STATE $N \Leftarrow N - 1$ +\ENDIF +\ENDWHILE +\end{algorithmic} +\end{algorithm} +\end{verbatim} +produces Algorithm~\ref{alg1} which is a slightly modified version of +the earlier algorithm for determining the value of a number taken to an +integer power. +In this case, provided the power may be negative provided the number is +not zero. + +The command \verb+\listofalgorithms+ may be used to produce a list +of algorithms as part of the table contents as shown at the beginning of +this document. +An auxiliary file with a suffix of \texttt{.loa} is produced when this +feature is used. + +\subsection{Options} + +The appearance of the typeset algorithm may be changed by use of the +options: \texttt{plain}, \texttt{boxed} or \texttt{ruled} during the +loading of the \texttt{algorithm} package. +The default option is \texttt{ruled}. + +The numbering of algorithms can be influenced by providing the name of +the document component within which numbering should be recommenced. +The legal values for this option are: \texttt{part}, \texttt{chapter}, +\texttt{section}, \texttt{subsection}, \texttt{subsubsection} +or \texttt{nothing}. +The default value is \texttt{nothing} which causes algorithms to be +numbered sequentially throughout the document. + +\subsection{Customization} + +In order to facilitate the use of this package with foreign languages, +methods have been provided to facilitate the necessary modifications. + +The title used in the caption within \texttt{algorithm} environment +can be set by use of the standard \verb+\floatname+ command which is +provided as part of the \texttt{float} package which was used to +implement this package. +For example, +\begin{verbatim} +\floatname{algorithm}{Procedure} +\end{verbatim} +would cause \textbf{Procedure} to be used instead of \textbf{Algorithm} +within the caption of algorithms. + +In a manner analogous to that available for the built in floating +environments, the heading used for the list of algorithms may be changed +by redefining the command \verb+listalgorithmname+. +The default definition for this command is +\begin{verbatim} +\newcommand{\listalgorithmname}{List of Algorithms} +\end{verbatim} + +\section{Labels and References in Algorithms} + +With the release of 2005-07-05, now \keyword{algorithmic} accepts labels +and references to specific lines of a given algorithm, so you don't have to +hardcode the line numbers yourself when trying to explain what the code +does in your texts. Thanks to Arnaud Legrand for the suggestion and patch +for this highly missed feature. + +An example of its use is shown in Algorithm~\ref{alg2}. +\begin{algorithm}[h!] +\caption{Calculate $y = x^n$} +\label{alg2} +\begin{algorithmic}[1] +\REQUIRE $n \geq 0 \vee x \neq 0$ +\ENSURE $y = x^n$ +\STATE $y \Leftarrow 1$ +\IF{$n < 0$} +\STATE $X \Leftarrow 1 / x$ +\STATE $N \Leftarrow -n$ +\ELSE +\STATE $X \Leftarrow x$ +\STATE $N \Leftarrow n$ +\ENDIF +\WHILE{$N \neq 0$} +\IF{$N$ is even}\label{alg:n-is-even} +\STATE $X \Leftarrow X \times X$ +\STATE $N \Leftarrow N / 2$ +\ELSE\label{alg:n-is-odd} +\STATE $y \Leftarrow y \times X$ +\STATE $N \Leftarrow N - 1$ +\ENDIF +\ENDWHILE +\end{algorithmic} +\end{algorithm} +See that, in line~\ref{alg:n-is-even}, we deal with the case of $N$ being +even, while, in line~\ref{alg:n-is-odd}, we give treatment to the case of +$N$ being odd. + +\end{document} |